2005 AMC 10A Problems/Problem 2

Problem

For each pair of real numbers $a$ $\neq$ $b$ , define the operation $\star$ as

$(a \star b) = \frac{a+b}{a-b}$ .

What is the value of $((1 \star 2) \star 3)$ ?

$\mathrm{(A) \ } -\frac{2}{3}\qquad \mathrm{(B) \ } -\frac{1}{5}\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \textrm{This\, value\, is\, not\, defined.}$

Solution

$((1 \star 2) \star 3) = \left(\left(\frac{1+2}{1-2}\right) \star 3\right) = (-3 \star 3) = \frac{-3+3}{-3-3} = 0 \Longrightarrow \mathrm{(C)}$

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2005 AMC 10A Problems/Problem 2

Problem

Solution

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